Express The Following Fraction In Simplest Form, Only Using Positive Exponents.${ \frac{15 W 6}{-3\left(w 4\right)^4} }$

Mar 8, 2025 by ADMIN 122 views

**Simplifying Fractions with Exponents: A Step-by-Step Guide**

Introduction

In mathematics, simplifying fractions with exponents is an essential skill that helps us express complex expressions in their simplest form. In this article, we will focus on simplifying the given fraction $\frac{15 w^6}{-3\left(w^4\right)^4}$ using only positive exponents.

Understanding Exponents

Before we dive into simplifying the fraction, let's quickly review what exponents are. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $w^4$ means $w$ is multiplied by itself four times, i.e., $w \times w \times w \times w$ .

Simplifying the Fraction

Now that we have a basic understanding of exponents, let's simplify the given fraction. We can start by simplifying the denominator using the rule $\left(w^4\right)^4 = w^{4 \times 4} = w^{16}$ .

\frac{15 w^6}{-3\left(w^4\right)^4} = \frac{15 w^6}{-3w^{16}}

Next, we can simplify the fraction by canceling out common factors in the numerator and denominator. We can start by dividing both the numerator and denominator by their greatest common factor (GCF), which is $3$ .

\frac{15 w^6}{-3w^{16}} = \frac{5 w^6}{-w^{16}}

Now, we can simplify the fraction further by using the rule $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is the base and $m$ and $n$ are the exponents.

\frac{5 w^6}{-w^{16}} = 5 w^{6-16} = 5 w^{-10}

Expressing Negative Exponents

Now that we have simplified the fraction to $5 w^{-10}$ , we need to express the negative exponent in terms of positive exponents. We can do this by using the rule $a^{-m} = \frac{1}{a^m}$ .

5 w^{-10} = \frac{5}{w^{10}}

Conclusion

In this article, we simplified the given fraction $\frac{15 w^6}{-3\left(w^4\right)^4}$ using only positive exponents. We started by simplifying the denominator using the rule $\left(w^4\right)^4 = w^{4 \times 4} = w^{16}$ . Then, we simplified the fraction by canceling out common factors in the numerator and denominator. Finally, we expressed the negative exponent in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .

Tips and Tricks

Here are some tips and tricks to help you simplify fractions with exponents:

Always start by simplifying the denominator using the rule $\left(a^m\right)^n = a^{m \times n}$ .
Next, simplify the fraction by canceling out common factors in the numerator and denominator.
Finally, express any negative exponents in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying fractions with exponents:

Don't forget to simplify the denominator using the rule $\left(a^m\right)^n = a^{m \times n}$ .
Don't cancel out common factors in the numerator and denominator without checking if they are actually common factors.
Don't forget to express any negative exponents in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .

Real-World Applications

Simplifying fractions with exponents has many real-world applications in fields such as physics, engineering, and computer science. For example, in physics, we often need to simplify complex expressions involving exponents to solve problems related to motion, energy, and momentum. In engineering, we may need to simplify expressions involving exponents to design and optimize systems such as electrical circuits and mechanical systems. In computer science, we may need to simplify expressions involving exponents to optimize algorithms and data structures.

Conclusion

Introduction

In our previous article, we explored the concept of simplifying fractions with exponents. We learned how to simplify fractions using positive exponents and express negative exponents in terms of positive exponents. In this article, we will answer some frequently asked questions (FAQs) related to simplifying fractions with exponents.

Q&A

Q: What is the rule for simplifying fractions with exponents?

A: The rule for simplifying fractions with exponents is to simplify the denominator using the rule $\left(a^m\right)^n = a^{m \times n}$ , then simplify the fraction by canceling out common factors in the numerator and denominator, and finally express any negative exponents in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .

Q: How do I simplify a fraction with a negative exponent in the denominator?

A: To simplify a fraction with a negative exponent in the denominator, you can use the rule $a^{-m} = \frac{1}{a^m}$ . For example, $\frac{1}{w^{-3}} = w^3$ .

Q: Can I simplify a fraction with a variable in the exponent?

A: Yes, you can simplify a fraction with a variable in the exponent. For example, $\frac{w^6}{w^4} = w^{6-4} = w^2$ .

Q: How do I simplify a fraction with a negative exponent in the numerator?

A: To simplify a fraction with a negative exponent in the numerator, you can use the rule $a^{-m} = \frac{1}{a^m}$ . For example, $\frac{w^{-3}}{w^2} = \frac{1}{w^3} \times w^2 = \frac{1}{w}$ .

Q: Can I simplify a fraction with a coefficient in the numerator or denominator?

A: Yes, you can simplify a fraction with a coefficient in the numerator or denominator. For example, $\frac{3w^6}{-2w^4} = \frac{3}{-2} \times \frac{w^6}{w^4} = -\frac{3}{2} \times w^{6-4} = -\frac{3}{2}w^2$ .

Q: How do I simplify a fraction with a negative coefficient in the numerator or denominator?

A: To simplify a fraction with a negative coefficient in the numerator or denominator, you can multiply the numerator and denominator by $-1$ . For example, $\frac{-3w^6}{2w^4} = -1 \times \frac{3w^6}{2w^4} = -\frac{3w^6}{2w^4}$ .

Q: Can I simplify a fraction with a variable in the numerator and a variable in the denominator?

A: Yes, you can simplify a fraction with a variable in the numerator and a variable in the denominator. For example, $\frac{w^6}{w^4} = w^{6-4} = w^2$ .

Q: How do I simplify a fraction with a variable in the numerator and a coefficient in the denominator?

A: To simplify a fraction with a variable in the numerator and a coefficient in the denominator, you can divide the numerator by the coefficient. For example, $\frac{3w^6}{w^4} = \frac{3}{1} \times \frac{w^6}{w^4} = 3 \times w^{6-4} = 3w^2$ .

Conclusion

In conclusion, simplifying fractions with exponents is an essential skill that helps us express complex expressions in their simplest form. By following the rules outlined in this article, we can simplify fractions with exponents and express negative exponents in terms of positive exponents. With practice and patience, you can become proficient in simplifying fractions with exponents and apply this skill to real-world problems in fields such as physics, engineering, and computer science.

Tips and Tricks

Here are some tips and tricks to help you simplify fractions with exponents:

Always start by simplifying the denominator using the rule $\left(a^m\right)^n = a^{m \times n}$ .
Next, simplify the fraction by canceling out common factors in the numerator and denominator.
Finally, express any negative exponents in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .
Use the rule $a^{-m} = \frac{1}{a^m}$ to simplify fractions with negative exponents.
Use the rule $\frac{a^m}{a^n} = a^{m-n}$ to simplify fractions with variables in the numerator and denominator.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying fractions with exponents:

Don't forget to simplify the denominator using the rule $\left(a^m\right)^n = a^{m \times n}$ .
Don't cancel out common factors in the numerator and denominator without checking if they are actually common factors.
Don't forget to express any negative exponents in terms of positive exponents using the rule $a^{-m} = \frac{1}{a^m}$ .
Don't multiply the numerator and denominator by $-1$ unnecessarily.